I don't think this makes sense. SMF is on MySQL and all users must then have MySQL. If AChat is going to be made independent of the system, that's fine. If not, I don't see the point in using two systems. I, for example, don't have SQLite. MySQL is also the general standard for the average user-- it can be used by more people.
One thing you could do to make it work would be to have some different prefix than SMF.
Personally I am probably not going to update my version of AChat after having customized it, so it won't hurt me if you do change it, but I think others would find MySQL useful also.